A Hamiltonian formulation of the BMS\(_4\)  symmetry on spacelike hypersurfaces fulfilling the above well-definedness requirement was developed in the papers [1]}, [2]}, [3]}. This was achieved through two distinct sets of boundary conditions. In [1]}, the parity conditions on the leading orders of the fields in an expansion at spatial infinity were taken to be different and inequivalent to the parity conditions of [5]}, even up to a coordinate transformation. In [2]}, [3]}, the parity conditions on the leading orders of the fields were taken to merely differ from those of [5]} by a coordinate transformation with specified fall-off. In both cases, the BMS\(_4\)  group emerges as asymptotic symmetry group of the theory. The first set of parity conditions (of [1]}) represents a more drastic twist of the parity conditions of [5]} because diffeomorphism invariant objects, such as the Weyl tensor, obey inequivalent conditions.
